Multiparametric Linear Programming Research Articles

Adaptive Power Management for Environmentally Powered Systems

Recently, there has been a substantial interest in the design of systems that receive their energy from regenerative sources such as solar cells. In contrast to approaches that minimize the power consumption subject to performance constraints, we are concerned with optimizing the performance of an application while respecting the limited and time-varying amount of available power. In this paper, we address power management of, e.g., wireless sensor nodes which receive their energy from solar cells. Based on a prediction of the future available energy, we adapt parameters of the application in order to maximize the utility in a long-term perspective. The paper presents a formal model of the corresponding optimization problem including constraints concerning buffer sizes, timing, and rates. Instead of solving the optimization problem online which may be prohibitively complex in terms of running time and energy consumption, we apply multiparametric programming to precompute the application parameters offline for different environmental conditions and system states. In order to guarantee sustainable operation, we propose a hierarchical software design which comprises a worst-case prediction of the incoming energy. As a further contribution, we suggest a new method for approximate multiparametric linear programming which substantially lowers the computational demand and memory requirement of the embedded software. Our approaches are evaluated using long-term measurements of solar energy in an outdoor environment.

Demand and generation cost uncertainty modelling in power system optimization studies

This paper describes the formulations and the solution algorithms developed to include uncertainties in the generation cost function and in the demand on DC OPF studies. The uncertainties are modelled by trapezoidal fuzzy numbers and the solution algorithms are based on multiparametric linear programming techniques. These models are a development of an initial formulation detailed in several publications co-authored by the second author of this paper. Now, we developed a more complete model and a more accurate solution algorithm in the sense that it is now possible to capture the widest possible range of values of the output variables reflecting both demand and generation cost uncertainties. On the other hand, when modelling simultaneously demand and generation cost uncertainties, we are representing in a more realistic way the volatility that is currently inherent to power systems. Finally, the paper includes a case study to illustrate the application of these models based on the IEEE 24 bus test system.

Process scheduling under uncertainty using multiparametric programming

AbstractIn this article, the problem of process scheduling under uncertainty was studied using multiparametric programming method. Based on the uncertainty type (prices, demands, and processing times), the scheduling formulation results in different parametric problems including multiparametric mixed integer linear (mpMILP), quadratic (mpMIQP), and general nonlinear programming (mpMINLP) problem. This article analyzes the solution characteristics and proposes a novel solution framework for specific mpMILP/mpMIQP problems addressing a wide variety of scheduling problems under different types of uncertainty, which are modeled as coefficient in objective function, as coefficient of integer variable and right hand side vector of the constraints. The main idea of the proposed framework is to decompose the problem into a series of smaller subproblems, each of them producing the parametric information around a given parameter value. The parametric solution of every subproblem is retrieved by solving a series of multiparametric linear programming (mpLP) and mixed integer linear/nonlinear programming problems (MILP/MINLP). Several examples were solved to analyze the complexity and the effectiveness of the proposed method. © 2007 American Institute of Chemical Engineers AIChE J, 2007

Lexicographic perturbation for multiparametric linear programming with applications to control

Optimal control problems for constrained linear systems with a linear cost can be posed as multiparametric linear programs (mpLPs) with a parameter in the cost, or equivalently the right-hand side of the constraints, and solved explicitly offline. Degeneracy occurs when the control input, or optimiser, is non-unique, which can cause chattering of the control input and overlap of the polyhedral regions of the explicit solution. This paper introduces a new and efficient approach to the computation of the solution to a degenerate mpLP with the parameter in the cost. Rather than solve the degenerate problem directly, we solve a lexicographically (symbolically) perturbed version of it that is guaranteed to be non-degenerate. We show that every optimal solution of the perturbed problem is an optimal solution to the original and that the perturbed solution is continuous, unique and defined over a set of non-overlapping polyhedral regions. Furthermore, we introduce a new method for computing the optimal solution in an adjacent region, which is very efficient in all cases and reduces to a single simplex pivot for non-degenerate regions. The proposed algorithm is particularly suited for the calculation of the explicit solution to a class of constrained optimal control problems, since it ensures that the control input is everywhere continuous and unique, thereby removing the danger of chattering in problems with linear costs. The algorithm is compared through example to existing proposals and a significant complexity improvement is demonstrated.

A New Methodology for the General Multiparametric Mixed-Integer Linear Programming (MILP) Problems

In this paper, a general algorithm is developed to address the multiparametric mixed-integer linear programming (mpMILP) problem with uncertain parameters in the left-hand side (LHS), right-hand side (RHS), and objective function coefficients simultaneously. The algorithm is based on a general multiparametric linear programming (mpLP) algorithm, which is derived using the optimality conditions of standard linear programming (LP) problem. The intent of the proposed framework is to propose a general framework to address different uncertainties in the process engineering problems. In addition, the paper also discusses the solution of the special problem where LHS uncertainty is included in the optimization model as a coefficient of continuous variable, and it notes the high computational complexity needed to retrieve the rigorous solution of large-scale problems, because of the nonlinearities of the objective function and nonconvex critical regions. Several numerical examples are presented to illustrate the effectiveness and applicability of the proposed method.

Multiparametric Linear Programming with Applications to Control

Parametric programming has received a lot of attention in the control literature in the past few years because model predictive controllers (MPC) can be posed in a parametric framework and hence pre-solved offline, resulting in a significant decrease in on-line computation effort. In this paper we survey recent work on parametric linear programming (pLP) from the point of view of the control engineer. We identify three types of algorithms, two arising from standard convex hull paradigms and one from a geometric intuition, and classify all currently proposed methods under these headings. Through this classification, we identify a third standard convex hull approach that offers significant potential for approximation of pLPs for the purpose of control. We present the resulting algorithm, based on the beneath/beyond paradigm, that computes low-complexity approximate controllers that guarantee stability and feasibility.

Constrained Optimal Control of Hybrid Systems With a Linear Performance Index

We consider the constrained finite and infinite time optimal control problem for the class of discrete-time linear hybrid systems. When a linear performance index is used the finite and infinite time optimal solution is a piecewise affine state feedback control law. In this paper, we present algorithms that compute the optimal solution to both problems in a computationally efficient manner and with guaranteed convergence and error bounds. Both algorithms combine a dynamic programming exploration strategy with multiparametric linear programming and basic polyhedral manipulation

Uncertainty analysis on the righthand side for MILP problems

AbstractA systematic framework is developed to solve the parametric mixed integer linear programming (pMILP) problems where uncertain parameters are present on the righthand side (RHS) of the constraints. For the case of multiple uncertain parameters, a new algorithm of multiparametric linear programming (mpLP) is proposed, which solves a number of nonlinear problems (NLP) iteratively. At each iteration, a point at which the objective value cannot be represented by the current optimal functions is found, and the new optimal function is included in the next iteration. Given the range of uncertain parameters in a MILP problem, the output of this proposed framework is a set of optimal integer solutions and their corresponding critical regions and optimal functions. A number of examples are presented to illustrate the applicabilities of the proposed approach and comparison with existing techniques. © 2006 American Institute of Chemical Engineers AIChE J, 2006

SHORT-TERM SCHEDULING OF CHEMICAL PROCESS INCLUDING UNCERTAINTY

This paper addresses the short-term scheduling of chemical process with uncertainty considerations. A multiobjective robust optimization method is proposed to identify Pareto optimal solutions, where Normal boundary intersection (NBI) technique is utilized in order to trace the Pareto optimal surface in the objective space, on which each point represents a trade-off between the various objectives. The issue is also addressed using parametric mixed integer linear programming (pMILP) analysis where uncertain parameters are present on the right hand side (RHS) of the constraints. For the case of multiple uncertain parameters, a new algorithm of multiparametric linear programming (mpLP) is proposed that does not require the construction of the LP tableaus but relies on the comparison between solutions at leaf nodes. Given the range of uncertain parameters, the output of this proposed framework is a set of optimal integer solutions and their corresponding critical regions and optimal functions.

A METHOD FOR OBTAINING CONTINUOUS SOLUTIONS TO MULTIPARAMETRIC LINEAR PROGRAMS

A modification of the geometric algorithm for solving multiparametric linear programs (mp-LP) is presented. The modification preserves the simplicity of the algorithm and ensures that the optimal, piecewise affine, mapping from parameter to solution space is continuous. When the mp-LP has non-unique solutions, the optimizer with the least Euclidian norm is selected.

Analysis of linear programming in model predictive control

The linear programming formulations of model predictive control are known to exhibit degenerate solution behavior. In this work, a multi-parametric linear programming technique is utilized to analyze the control laws that are generated from various linear programming based MPC routines. These various routines explore a number of factors, including objective function selection and constraint handling on the control laws generated from LP based MPC. A single input single output system is used to demonstrate that the use of input velocity penalties, input blocking, and ∞ -norm objective functions can limit or eliminate this undesirable behavior. Finally, a paper machine cross directional control problem is used to demonstrate the control laws generated from LP based MPC for a multivariable example.

Geometric Algorithm for Multiparametric Linear Programming

We propose a novel algorithm for solving multiparametric linear programming problems. Rather than visiting different bases of the associated LP tableau, we follow a geometric approach based on the direct exploration of the parameter space. The resulting algorithm has computational advantages, namely the simplicity of its implementation in a recursive form and an efficient handling of primal and dual degeneracy. Illustrative examples describe the approach throughout the paper. The algorithm is used to solve finite-time constrained optimal control problems for discrete-time linear dynamical systems.

Min-max control of constrained uncertain discrete-time linear systems

For discrete-time uncertain linear systems with constraints on inputs and states, we develop an approach to determine state feedback controllers based on a min-max control formulation. Robustness is achieved against additive norm-bounded input disturbances and/or polyhedral parametric uncertainties in the state-space matrices. We show that the finite-horizon robust optimal control law is a continuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric linear programs. When the optimal control law is implemented in a receding horizon scheme, only a piecewise affine function needs to be evaluated on line at each time step. The technique computes the robust optimal feedback controller for a rather general class of systems with modest computational effort without needing to resort to gridding of the state-space.

An Algorithm for the Solution of Multiparametric Mixed Integer Linear Programming Problems

In this paper, we present an algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems involving (i) 0-1 integer variables, and, (ii) more than one parameter, bounded between lower and upper bounds, present on the right hand side (RHS) of constraints. The solution is approached by decomposing the mp-MILP into two subproblems and then iterating between them. The first subproblem is obtained by fixing integer variables, resulting in a multiparametric linear programming (mp-LP) problem, whereas the second subproblem is formulated as a mixed integer linear programming (MILP) problem by relaxing the parameters as variables.

Simultaneous incorporation of flexibility and economic risk in operational planning under uncertainty

In this paper, an integrated metric is introduced with which future plan feasibility together with the potential economic risk is assessed for two-period linear planning models. It is based on the concepts of flexibility, the ability of handling uncertainty while meeting production requirements, and maximum regret, the maximum potential economic loss involved in the implementation of a plan in the face of uncertainty. An algorithmic procedure is proposed for the estimation of such a combined metric. It involves the solution of two multiparametric linear programming subproblems for the evaluation of maximum regret, with which analytical expressions of the regret as a function of the uncertain parameters and the plan are obtained. These expressions are then incorporated in a mixed-integer index programming formulation for the evaluation of a combined flexibility-regret index. In this way, trade-offs between plan flexibility and economics are established. The incorporation of these analytical tools into an overall planning framework is also shown and illustrated with example problems.

Multiparametric linear fractional functionals programming

In this paper a multiparametric linear fractional functionals program, with parameters appearing only in the objective function, is generated. The optimum solution of this parametric program is supposed to satisfy the constraints as equations only. It is also shown that the set of parameters forms a convex polyhedron.

Validity of Local Search in Multiparametric and Vector Linear Programming

Validity of Local Search in Multiparametric and Vector Linear Programming

Validity of Local Search in Multiparametric and Vector Linear Programming

Validity of Local Search in Multiparametric and Vector Linear Programming

目的関数の係数に複数個のパラメータをもつ0-1混合整数計画問題

While continuous parametric analysis is of some interest in integer linear programming, it is desirable to deal with parametric integer programming problems which include multiple parameters for the right-hand sides or objective function coefficients in the model. Because, by the introduction of multiple parameters into the integer linear model, the interdependences among the entire system, its environment and the subsystems can be varied. Thus, multipara-metric programming can become an important tool for solving various problems arising in design and/or control of real life engineering systems.Although several literature on parametric integer programming (PIP) with single parameter for the right-hand sides or objective function coefficients have been published, no work has been done on PIP with multiple parameters in the model.The purpose of this paper is to present an effective method for finding optimal solution of parametric mixed integer linear programming with multiple parameters for the objective function coefficients. The present method is an extension of the branch-and-bound algo-rithm for 0-1 parametric mixed integer programming with single parameter. In order to solve a subproblem defined on each node of the branch-and-bound tree, a relaxation problem is defined, which is formulated as a multiparametric linear programming. For each node, a range set of multiple parameters is defined and it must be convex. Otherwise, thealgorithm for solving the multiparametric linear programming cannot be applied. To construct the convex set of parameters for each node, parameter regions obtained by solving the multiparametric linear programming are effectively used. To show the effectiveness of the present method, a factility location problem is considered, where the objective function coefficients are varied continuously by the ntroduction of multiple parameters.

The max-flow problem with parametric capacities

In this paper we present a comprehensive analysis of the max-flow problem with n parametric capacities, and give the basis for an algorithm to solve it. In particular we give a method for finding the max-flow value as a function of the parameters, and max-flows for all parameter points, in terms of max-flow values to problems at certain key parameter points. In the problem with nonzero lower bounds on the arc flows, we derive a set of linear constraints whose solution set is identical to the set of all feasible parameter points. The intrinsic difficulty of the problem is compared with that of the general multiparametric linear programming problem, and thus light is shed on the difficulty of the latter problem, whose complexity is currently unknown.